BSTJ 59: 10. December 1980: Impairment Mechanisms for SSB Mobile Communications at UHF with Pilot-Based Doppler/Fading Correction. (Leland, K.W.; Sollenberger, N.R.)

Copyright © 1980 American Telephone and Telegraph Company

The Bell System Technical Journal

Vol. 59, No. 10, December 1980

Printed in U.SA.

Impairment Mechanisms for SSB Mobile

Communications at UHF with Pilot-Based

Doppler/Fading Correction

By K. W. LELAND and N. R. SOLLENBERGER
(Manuscript received April 21, 1980)

Several impairments are associated with the pilot- based correction
of a single sideband (ssbJ voice signal for the effects of vehicle motion
in a Rayleigh multipath field. These include the distortion caused by
a limit on the available correction gain, the distortion resulting from
frequency selective fading, the degradation of the signal-to-interfer-
ence (s/\) [noise (s/n)] ratio, and the distortion induced because of
interference to the pilot from a cochannel pilot or from noise. We
have investigated these impairments for a specific feedforward,
phase- and gain- correcting, ideal receiver. For this receiver, the s/i
ratio is degraded at baseband because of the time-varying nature of
the gain correction signal and the assumption that the correction
signal is correlated with the desired voice signal fading envelope but
is independent of the interferer fading envelope. A limit on correction
gain appears necessary to limit the magnitude of several of the
impairments, and has been considered as a parameter in the analysis
of each impairment type. The results are here extended to character-
ize an "equal-gain" space diversity receiver. Diversity combining is
applied after phase correction, but before gain correction to maintain
proper voice correction while retaining the benefits of the improved
envelope statistics. This paper deals only with the SSB idealized
channel, not with a system employing ssb modulation; the authors
recognize that many complicated systems problems (e.g., antenna
combining, channel generation, and cost reduction) must be solved
before a working system using the channel analyzed herein can
become a reality.

I. INTRODUCTION

Conventional suppressed carrier single sideband (ssb) uhf mobile
communication links exhibit severe voice-quality impairments because

1923

of the phase and amplitude modulation associated with motion in a
multipath field. In view of this, various voice correction methods have
been occasionally considered over the years. 1,2 These include the
technique of incorporating a separate pilot rf frequency at constant
level to provide a means of correcting the voice information at the
receiver. This idea is now receiving increased attention, since re-
evaluation of ssb for providing land-mobile radio communication. 3
Although pilot-based correction is now often presented as a probable
solution to the fading-induced voice quality problem, 4-6 an analysis of
the residual impairments and the performance of a pilot-based ssb
channel in the presence of cochannel interference has not been given.
Such an analysis is the object of this paper. The analysis is based on
a specific receiver configuration employing ideal elements. This paper
deals only with the ssb idealized channel, not with a system employing
ssb modulation; the authors recognize that many complicated systems
problems (e.g., antenna combining, channel generation, and cost re-
duction) must be solved before a working system using the channel
analyzed herein can become a reality.

The intuitive expectations for pilot-based operation, that stem from
considerations of suppressed-carrier, pilotless, ssb operation, are often
incorrect. Such is the case with the plausible and common assumption
that the signal-to-interference (s/i) [or signal-to-noise (s/n)] ratio at
baseband is the same as the corresponding ratio at rf. In fact, we will
show that the s/i (s/n) ratio of a pilot-based system is deteriorated at
baseband in the presence of either flat or frequency-selective Rayleigh
fading, even when cochannel pilot interference is absent. This can be
thought of as the ssb signal-to-interference "disadvantage," analogous
to the textbook fm s/n advantage for fm operation in the nonfading
environment. The deterioration of the s/i at baseband is caused by
the time-varying nature of the gain-correction signal and the assump-
tion that the gain correction signal is correlated with the desired voice-
signal fading envelope but is independent of the interferer-fading
envelope. Although effects such as this may be established empirically
through system trials, an understanding of their basis is desirable for
the design of efficient ssb systems and for the most realistic comparison
of different, and as yet untested, "paper" systems.

The effect of a limit on the gain available for fast-gain correction is
considered in Section II, as in a previous analysis of gain correction for
am mobile radio by M.J. Gans. 2 While some limit is a consequence of
practical hardware, a limit is actually required to bound impairments
in the presence of frequency-selective fading, as well as to bound the
s/i disadvantage. The tradeoff is essentially between residual-correc-
tion distortion in the absence of delay spread and the distortion in the
presence of significant (3 /us) delay spread. The effect on the s/i
disadvantage also influences the choice of a correction-gain limit.

1924 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1 980

Frequency-flat Rayleigh fading was assumed in the analysis of residual-
gain limit distortion and s/i disadvantage. A frequency-selective
model 7 was assumed for the analysis of delay-spread effects on the
correction process. In Section III, iV-branch space diversity was applied
to the correcting receiver and the effects on the above impairments
were analyzed; the impairments in this latter case were found to be
bounded even when no limit is placed on the available correction gain.

II. RECEIVER ANALYSIS

Figure 1 diagrams the idealized pilot-based receiver under consid-
eration. Both gain and phase correction are included because both
have been experimentally found to be necessary to provide a natural
rendition of voice information. The voice-channel selective filtering is
placed before the phase and gain correction stages to avoid introducing
inband adjacent channel interference, resulting from the complex
multiplication of the incoming RF by a relatively wideband signal
which is likely to be uncorrelated with any adjacent channel signals
that are present. This filter placement requires the voice filter to be
delay equalized across its passband. In any case, the absolute delay of
the voice and the pilot paths must be matched prior to correction in
phase and amplitude. In practice, a slow age would be included before
the receiver elements in Fig. 1, with the intention of removing the
slow, log-normally distributed, shadow fading that is imposed on the
rapidly fading signal. The age would receive a control signal from the
pilot envelope detector and adjust the gain to keep the average power
in the pilot channel constant. An age time constant of 0.1 to 1.0 s seems
appropriate for this purpose. In the following sections, we assume that
this slow shadow fading can be successfully removed, and consequently
we will not consider it in the analysis.

Amplitude companding is applicable to a pilot-based ssb system

r

SLOW
AGC

PILOT

BPF

(Translating)

ENV
DET

VT

¥

'

LIMITER

AUDIO
BPF

b

VOICE

—

PHASE

a

LIMITED GAIN

C

BPF

C

ORR

CTIOK

:orre

CTION

c = a • MIN (1/b. 1/£b )
Fig. 1 — Pilot-based Doppler/fading correction.

SSB VOICE SIGNAL IMPAIRMENT MECHANISMS 1925

because of the constant-gain nature of the corrected channel. As is
customary, amplitude companding is expected to improve the noise
and interference (cross talk) performance of the system. The purpose
of this paper is to analyze the basic channel, to which amplitude
companding can be added to improve the apparent performance by
reducing the impairment level at the receiver output when the desired
speech is absent.

2. 1 Correction distortion

Residual correction distortion in a flat-fading environment is caused
by the system's inability to maintain constant channel gain during the
deepest fades caused by the imposed correction-gain limit. The correc-
tion distortion limits the best-case performance of the mobile com-
munication link in a noiseless, interference-free, flat-fading environ-
ment. This impairment is termed distortion because the average im-
pairment power is proportional to the average desired speech power.
However, it is not harmonically related to the speech. Also note that
after correction, a small amount of distortion energy falls outside the
original voiceband and is removed by the final audio filter. This is
neglected in this analysis.

The original voice signal can be conveniently represented as

s(t) = a(t)e m \ (1)

where a(t) and 6(t) are audio envelope and phase functions, respec-
tively. After ssb modulation and transmission, this signal and its pilot
are received as

s r (t) = a(t)r(t)exp[i(u) c t + 6{t) + (j>(t))~\ + r{t)exp[i(w p t + $(*))]

= z(t)a(t)exp[i(u c t + S(t))] + z(t)exp(iu p t), (2)

where z(t) = r(£)e** (/) is the complex fading envelope caused by vehicle
motion. Reference 7 describes the model of propagation in the mobile
environment under consideration. In this standard model, z(t) is a
complex random process, the real and imaginary parts of which can be
taken as independent real Gaussian processes with identical symmetric
power spectra. For the E field vertical-whip antenna, these power
spectra, and hence the power spectrum of z(t), are of the form

1-1/2

z</)-|i-l-H , 0)

-I-©-

where fd is the maximum Doppler frequency based on the vehicle
speed. Figure 2 shows this spectrum. The above model possesses the
widely used Rayleigh statistics for r(t), because of the independent
Gaussian real and complex parts of z(t). In the flat-fading or negligible
delay-spread environment, z(t) multiplies the original signal compo-

1926 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1 980

-»m +f m

Fig. 2— Pilot spectrum for an E field antenna.

nents independent of their frequency, as in (2). This is assumed for
this section. Including significant delay spread yields frequency selec-
tive fading and results in the complex fading envelope at one frequency
having only partial correlation with the fading envelope at a different
frequency. Finally, the model is ergodic; consequently, throughout this
paper, averages over time, which are denoted with a bar, are used
interchangeably with ensemble averages.

To develop a basis for voice correction, the received pilot, repre-
sented by the second term in (2), is separated from the voice by
filtering. The ideal filter bandwidth required is twice the Doppler shift
associated with the highest expected vehicle speed, for example, 160
Hz at 60 mph for 900-MHz center frequency. The absolute delay in
both paths is assumed to be equalized, to allow timely amplitude and
phase correction. If the delay is equalized, the magnitude of the delay
does not affect the following calculations and hence will not be carried
along. Note also, that the pilot signal is only delayed in passing through
the pilot filter, because of the assumed existence of delay equalization
across the filter passband and the assumption that the filter bandwidth
is chosen wide enough to encompass the pilot spectrum. After multi-
plying by the hard limited pilot and subsequent lowpass filtering to
remove the sum component, the phase corrected output is written as

s p (t) = a(t)r(t)e

iBU)

(4)

The amplitude correction is gain limited to follow fades only as deep
as er , where ro is the rms average of r(t), and e is the correction-gain
limit parameter. The maximum available correction gain is thus de-

SSB VOICE SIGNAL IMPAIRMENT MECHANISMS 1927

fined as 1/e. The pilot envelope r(t) is detected in the receiver and
used by the gain correction block to develop the final corrected audio
output given by

s b (t) = r(t)mm(-^-, — )a(t)e mt \ (5)

where min( • , • ) is defined as the smaller of the two arguments on an
instantaneous basis. The signal-to-correction distortion (s/d) ratio is
then defined as

sdr = s 2 /(s b - s) 2 . (6)

Substituting (1) and (5) into (6) and simplifying, based on the inde-
pendence of voice and fading envelope signals, gives the correction
distortion in the flat Rayleigh-fading environment as

sdr =

2 - 1

r(r,iml1 ^)'^)- 1

(7)

The fading envelope is assumed to be characterized accurately by the
Rayleigh density function:

p(r)=%exp(-r 2 /r 2 )u(r). (8)

To

The s/d ratio may now be computed as

sdr =

rte-o's

exp(-r 2 /ro) dr

(9)

This reduces to

1 .. _., . . « Z (-DV n

sdr = (4 (1 ~ e~*) + * ~ 2 S /o ''■',! i ) • (10)

\e 2 n-o (2/i + l)n!/

an expression for the best-case distortion for the moving pilot-based
receiver with a fixed available gain limit. Figure 3 gives a plot of this
expression as a function of 1/e, the available correction gain.

The correction process has been simulated by a digital computer
program to supplement and check the analytical results. The flat-
fading, no-interference version of the simulation is based on (3), (5),
and (8). The corrected output and the time average of the power
spectrum of the corrected output were obtained from the simulation
and are given in Figs. 4 and 5. This trial corresponds to a frequency of
900 MHz in a flat Rayleigh-fading environment, and a vehicle speed of
70 mph. The shape is characteristic of distortion spectra produced at
a wide range of correction gain limits and with or without simulated

1 928 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1 980

DU

/ TWO-BRANCH
/ DIVERSITY **

40

CO

_l
m

/ FOUR-BRANCH
/ DIVERSITY /

CO

o
III

a

Z 30

■ / /

z
o

/ /

^ NO DIVERSITY

1-

DC

o

fe 20
D

J / /

-I
<
Z

55

y\^^

10

■

i i

I I I I

10 15 20 25

CORRECTION GAIN LIMIT IN DECIBELS

Fig. 3 — Gain-limit distortion (flat fading).

30

35

urban delay spread (discussed later). Of course, these factors do
directly affect the amount of correction distortion.

2.2 Effects of correction on interference

The baseband s/i ratio (sir ) will be expressed as a function of the
cochannel voice signal-to-interference ratio at RF(siR r ) and the correc-
tion gain limit. The pilot signal is assumed to be free of interference,
including cochannel pilot interference (which will be discussed in
Section 2.4). The receiver input in the case of a desired signal plus a
single interferer is given by

s r (t) = a d r d exp[i(u c t + d + <M] + r d exp[i(cj pd t + <f> d )]

+ a[a,r ( exp[i(w c £ + ft + <M] + nexp[i(o3 pi t + <f>,)]], (11)

where

a d , at are the desired and interfering voice envelopes,
d , 0i are the desired and interfering voice phase functions,
o)c is the voice virtual carrier frequency,
copd is the desired pilot frequency,

u)pi is the interfering pilot frequency — assumed for this analysis to
be offset from the desired voice and pilot,
r d , r, are the Rayleigh fading envelopes,

SSB VOICE SIGNAL IMPAIRMENT MECHANISMS 1929

I |

1 |

-2

£ -4

CJ
LU

Q

Z

a.

1-

8 -6

uj

a.
O

-I

UJ

>
Z

UJ

III

I" 8

-10

-12

I I I

I

0.2

0.4 0.6

TIME IN SECONDS

0.8

Fig. 4 — Tone envelope at the corrected output, 20-dB correction.

fa, 4>i are the rapid phase- variation functions due to motion,
a is a relative propagation constant that determines the s/i at rf
(i.e., siR r = 1/a 2 ).

The corrected receiver output can then be written as *

s b it) = s(t) + d(t) + i(t),

where

s(t) = a d e i0d ,
d(t) =

1 l i ■
rd nun| — , I — 1

Td €Tdo,

ade

id,i

(12)

(13)
(14)

ilt) = an mini — , )a,exp[i(^ + <fc - </>d)]. (15)

\r d er d o/

1 930 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1 980

The signal-to-correction distortion ratio remains defined as before.
The signal-to-interference ratio at the output is defined conventionally

as

SIR*, =

s(ty

(16)

For assumed equal power averages for a d {t) and at(t), and indepen-
dence of the voice and fading envelopes, (13) and (15) can be substi-
tuted into (16) and the result reduced to

or

SIR 6

= rf mini — , siR r

| \rd er do J

SIRt = y 'SIR r ,

(17)
(18)

Q n —

< -10 -

-500

FREQUENCY IN HERTZ

Fig. 5 — Tone-power spectrum at corrected output, 20-dB correction flat fading.

SSB VOICE SIGNAL IMPAIRMENT MECHANISMS 1931

where y is the signal-to-interference disadvantage ratio (siR r to sir*,).
The expression for y can be found by performing the indicated aver-
ages. This gives

r = f rtfiton) *(^ [" jfe) dr* * £ g$ *-) , (19)

where

p(r t ) =p(r d ) = 2 -5 exp(-r 2 /ro)u(r).

To

(20)

Using the known series expansion for the last integral above, 8 the
expression can be rewritten as

1 , °° (-l)V

y = -2 ln(e) - f + -, (1 - e-<) - I . , (21)

€ n-i nni

where \p is the Euler constant (0.5772' • • ).

The above series converges rapidly for e < 1, the region of practical
interest. Figure 6 gives the s/i disadvantage, y, for the pilot-based
receiver under consideration, as a function of available correction gain.
The disadvantage varies from 6 to 8 dB for the range of gain limit
from 15 dB to 25 dB. An analysis of the s/i performance of fm in the

TWO-BRANCH DIVERSITY

FOUR-BRANCH DIVERSITY

10 20

CORRECTION GAIN LIMIT IN DECIBELS

30

Fig. 6 — Signal/interference disadvantage.
1932 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1 980

flat-fading environment can be taken from Jakes. 2 However, final
conclusions from modulation comparisons must also await the experi-
mental comparison of the subjective nature of the two different types
of impairments. In addition, other factors, such as talker activity, are
expected to have an unequal impact on the performance of different
systems and other important impairments may be present due to
particular hardware implementations.

The correction effect on the signal-to-noise ratio is identical to the
effect on interference, described above, as long as the noise and fading
processes are assumed independent.

2.3 Delay spread

The existence of a spread in arrival time of the various multipath
components leads to the frequency selective fading phenomenon, for
vehicles moving in the multipath field. In this case, the correlation in
phase or amplitude between two pilots separated in frequency and
transmitted from the same antenna is high for small frequency sepa-
ration and falls to essentially zero as the separation substantially
exceeds the correlation bandwidth. The correlation bandwidth is the
frequency separation for a given environment, at which the complex
correlation coefficient for the two pilot phasors has dropped to a given
value (0.5 is common and is used here). The correlation bandwidth
encountered in the mobile environment is quite large compared to 3
kHz, i.e., 90 kHz or more, over 99 percent of the urban area.f However,
as was pointed out by Gans 2 for the am case, the gain-correction
process requires a very high degree of correlation between the rapid
phase and gain variation imposed on the pilot to that of the phase and
gain variation imposed on the voice band, if voice distortion is kept
small. The relationship between the rf angular frequency separation
(s), the delay spread (a), and the complex envelope correlation (X), is
given exactly 2 for exponential delay distributions and approximately
for typical distributions, as

X 2 = (1 + sV)-\ (22)

From this we can see that for a pilot to voice-tone separation of 2 kHz
and 3 jus urban delay spread, the pilot to tone correlation is 0.9993.
This is thought to be a near worst case condition for the urban
environment (1 percentile probability), based on the limited delay-
spread measurements reported in Ref. 9. The audio-frequency depen-
dent distortion resulting from the effect of pilot- voice band decorrela-
tion can be evaluated from the signal equations for the receiver:

Audio: s(t) = a(t)e m \ (23)

f Based on a one percentile 3 /is rms delay spread from Ref. 9 and eq. (22).

SSB VOICE SIGNAL IMPAIRMENT MECHANISMS 1933

Received Input:

s r (t) = ar v exp[i{o3 c t + 6 + </>„)] + r p exp[i(w p * + </> p )]

= z v a exp[i(u c t + 0)] + z P exp(iu p £), (24)

Received Output:

s b (t) = arjoaml — , — )exp[i(0 + <t> v - <ML ( 25 )

\r p a-poj

where

z v = r v (t)e^ lt) and z p = r p (t)e*> lt)

are correlated complex Gaussian random processes, band limited to
twice the vehicle speed Doppler frequency. The spectrum assumed for
these processes is appropriate for a vertically-polarized E field antenna
as described in Section 2.1. The signal-to-correction distortion ratio is
written as before as

SDR = \*B)

(s b (t) - s(t)) 2

After substituting (23) and (25) into the above and performing the
indicated averages, the sdr can be written in terms of the envelope
and phase properties as

l-i-

SDR =

r„mm(± f — W*M
\r p crpoj

-i

(27)

where the subscripts p and v denote pilot and voice, and the subscript
o indicates the rms time average. The above process, in conjunction
with (22), has been simulated on the digital computer using Monte
Carlo techniques. The results, which again depend on the gain avail-
able for correction, as well as the audio-pilot frequency separation, are
shown in Fig. 7 for the l-/xs urban delay-spread environment and in
Fig. 8 for the 3-jus environment. Frequency separations of 2 and 3 kHz
were used. The 2-kHz separation case was chosen to give distortion
results representative of what might be expected for a 300- to 3000-Hz
voice channel employing a pilot at or near the virtual voice-carrier
position. The 3-kHz separation gives an upper bound to the distortion
in the above case. Figure 9 shows the dependence of distortion on
delay spread at a fixed 20-dB correction gain and a 2-kHz frequency
separation.

The results indicate that the distortion introduced by delay spread
will be relatively modest (typically sdr > 20 dB), and quite limited in
occurrence for a particular coverage area, although an experimental

1 934 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1 980

FOUR-BRANCH, S = 2 KHZ

TWO-BRANCH, S = 2 KHZ

.

10 20

CORRECTION GAIN LIMIT IN DECIBELS

30

Fig. 7— Delay-spread distortion. N-branch diversity-simulation results. Delay = 1 /is,

S = frequency separation.

FOUR-BRANCH, S = 2 KHZ
FOUR-BRANCH, S = 3 KHZ

TWO-BRANCH, S = 2 KHZ

TWO-BRANCH, S = 3 KHZ

ONE-BRANCH, S = 2 KHZ
ONE-BRANCH, S = 3 KHZ

10 20

CORRECTION GAIN LIMIT IN DECIBELS

30

Fig. 8— Delay-spread distortion. N-branch diversity-simulation results. Delay = 3 (is,
S = frequency separation.

SSB VOICE SIGNAL IMPAIRMENT MECHANISMS 1935

10

1 2

DELAY SPREAD IN MICROSECONDS

Fig. 9 — Delay-spread distortion. Correction limit = 20-dB, 2-kHz frequency separation.

evaluation must be made to determine the subjective nature of this
impairment. The audio tone output power spectrum for the 3-/is delay
spread, 2-kH separation, case is given in Fig. 10 and is similar in shape
to the output spectrum for gain-limit distortion alone (Fig. 5).

In reviewing this paper, M. J. Gans gave a closed-form expression
for (27) as

^•+(1-X 8 )£ 1 (€ 2 )- — erf(c)

SDR =11+

1-e

(27')

The simulation results check well with this exact expression.

2.4 Cochannel pilot interference

The interference effect from a cochannel pilot on the desired channel
pilot decorrelates the received pilot and voice-complex envelopes.
Thus cochannel pilot interference induces correction distortion in a
manner very similar to that induced by operation in a high delay
spread environment. Pilot-interference distortion results are obtained
by simulating the pilot envelope z p {t) as the sum of an independent
interfering pilot and the desired pilot. Frequency flat fading is assumed,
so consequently the desired pilot envelope is taken as identical to the
voice envelope z v (t). The processes z p (t) and z v (t) are then used in
conjunction with (25) to numerically estimate the distortion due to
cochannel pilot interference. Figure 11 presents the results. The dis-
tortion induced by cochannel pilot interference is seen to be approxi-

1 936 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1 980

mately the same level as the baseband voice interference over the
most likely operating conditions, but is present only during the desired
speech.

III. IMPAIRMENTS OF A SPACE DIVERSITY RECEIVER

The three impairments discussed above for the nondiversity receiver
are heavily dependent on the fading statistics of the complex fading
envelope. Consequently, it is reasonable to expect that diversity sig-
nificantly improves the situation. Indeed, this has been found to be
the case. Figure 12 gives the equal-gain N-branch diversity receiver
assumed. The individual branch signals are first phase corrected and
then summed. The envelope correction signal is formed from the sum
of the individual branch-pilot envelopes. Amplitude correction is then
applied exactly as in the nondiversity case. Diversity combination in

ui _io -

-30

-500

FREQUENCY IN HERTZ

Fig. 10— Tone-power spectrum at corrected output. 20-dB correction, 3-/is delay
spread, 2-kHz frequency separation.

SSB VOICE SIGNAL IMPAIRMENT MECHANISMS 1937

s

o 20

Q

TWO-BRANCH DIVERSITY

Z

o

1-

O 10

NO DIVERSITY

H

^

<

a

I

1

10 20

PILOTS/I IN DECIBELS

30

Fig. 11 — Pilot-interference distortion. Maximum correction = 20 dB.

this way maintains the necessary voice correction and simultaneously
achieves the benefits of having improved signal-envelope statistics
prior to the gain correction step.

The simulation program has been extended to include iV-branch
diversity based on the development in Sections 3.1 to 3.3. Results for
equal gain combining will be reported here. Analytical results have

N IF BRANCH

N BRANCHES (ENV)

SLOW
AGC

PILOT
BPF

ENV
DET

ill

VOICE
BPF

PHASE
CORRECTION

LIMITED

GAIN

CORRECTION

PHASE
Z, CORRECTED
AUDIO

ITT

AUDIO
PROCESSING

c = a-MIN (1/b, 1/£b )

N BRANCHES (AUDIO)

Fig. 12 — N-branch diversity-pilot-based receiver.

1 938 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1 980

been obtained for the signal-to-interference disadvantage for the two-
branch diversity case. The fading statistics on each diversity branch
were taken to be independent, and as described in (20).

3.1 Gain limit distortion— diversity

The signal equations for the diversity receiver in the frequency flat-
fading environment are given by

Received Input(nth branch):

Sm(t) = ar n exp[i(u c t + 6 + <?>„)] + r n exp[i(w p t + </>„)], (28)

Corrected Output: s b (t ) = r 8 min( — , \a*r*, (29)

where the subscript s denotes a summation over iV-diversity branches,
and the subscript o denotes the rms time average. The distortion as
defined in (6) can be evaluated from (1) and (29) for the iV-branch
diversity case as

-i

(30)

SDR =

r s min — , — 1

The fading envelopes were assumed to be independent and Rayleigh
as given in (8). Results of the simulation based on (8) and (29) are
given in Fig. 3 for two- and four-branch diversity receivers.

3.2 Signal to interference disadvantage — diversity

The signal-to-interference disadvantage as defined in (18) can be
evaluated for the iV-branch, equal gain, diversity receiver following the
previous development of (17) and including diversity as in Section 3.1.
This leads to the following expression for s/i disadvantage:

■ ( l l \
un — , ,

\rda &dso/

= Nrlmm{ — , , (31)

where r„ is the rms average common to each branch envelope. The
subscripts d and i denote the desired and interferer envelopes and the
other subscripts are defined in Section 3.1. Signal-to-interference dis-
advantage results based on the numerical evaluation of (31) are given
in Fig. 6. Rayleigh envelope statistics, as given in (20), were again
assumed.

For the two-branch case, the probability density function of the sum
of two Rayleigh envelopes can be obtained by a series evaluation of
the convolution integral involving the individual density functions.
This can then be applied to (31) and the indicated averages can be
performed to give a series expression for the s/i disadvantage for two-

SSB VOICE SIGNAL IMPAIRMENT MECHANISMS 1939

branch equal-gain diversity as

„_o n!2

^2^)[i(^ eH&,2/2G ^- )

- 2 7^e- lSt)2/2 G{n + 1, m) + ^r^ Gin i l , 71 + J )

^0 (Se) 2 -
where 5 - r so /r = V2 + n/2 and

G(/i, m) =

(W

n\r

(n — m)\'

(32)

(33)

In the limit of infinite available correction gain, the expression for two-
branch diversity s/i disadvantage reduces to a constant given by

y = 2 I (-l) n "

n=0

v 2« + 1 2ra + 3
= 1.142 or 0.58 dB.

(34)

Here again, a closed-form expression has been supplied, in review,
by M. J. Gans for (32), which gives the s/i disadvantage as

y = ^—^ — e^erf (x) + w[l - erf 2 (x)], (32')

where x = eVl + it/4.

3.3 Delay spread — diversity

Following the analysis of Section 2.3, the diversity signal-to-distor-
tion ratio at the receiver output is found to be

SDR =

nun

er„

2 r lw e«*"-*» ) - 1

(35)

where the subscripts p, v, s, and o are as previously defined, and the
subscript n denotes the diversity branch number. This expression can
be shown to be finite (nonzero) in the case of unlimited available
correction gain, completely uncorrelated pilot and voice envelopes,
and two-branch diversity, by applying the series expansion for the
probability density of r s in the indicated average and replacing the
complex and signed quantities with their respective magnitudes. The
complex envelopes of one branch were taken as independent of the
complex envelopes of other branches. As in Section 2.3, the complex
pilot envelope is assumed to be correlated with the complex fading
voice envelope, corresponding to a given delay spread. For the assumed
exponential delay spread profile, the relationship is given in (22).
Based on these assumptions, the process has been simulated for the

1 940 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1 980

same conditions used to produce the nondiversity delay spread results.
The diversity delay spread results are given in Figs. 7 and 8.

IV. CONCLUSIONS

This paper has addressed the pilot-based correction of ssb voice
signals for the effects of vehicle motion in a Rayleigh-distributed
multipath field. Results were presented for the specific, idealized
receiver in Fig. 1 which employs feed-forward gain and phase correc-
tion. The amount of gain available for correction in the receiver is an
important parameter that affects the audio quality of a pilot-based ssb
receiver in the fading environment. This gain limit is bounded on the
low end by the distortion that results from imperfectly correcting the
voice signal during fades. The gain limit is bounded on the high end by
the increasingly detrimental effects of correction on cochannel inter-
ference and by urban delay spread. Approximately 20 dB of correction
relative to the rms envelope appears reasonable, for a nondiversity
receiver. For this degree of correction, the residual distortion is 27 dB
below the speech, in the absence of significant delay spread.

The correction process also deteriorates the s/i (s/n) ratio at base-
band by 5 to 9 dB compared to the sir (snr) at rf for the range of
useful correction gain limits of 10 to 30 dB. This s/i disadvantage is
about 7 dB for the 20-dB correction limit mentioned above. Cochannel
pilot interference would additionally contribute an approximately
equal impairment level while the desired speech is present (pilot
interference distortion).

The effects of urban delay spread do not appear to represent a major
obstacle. The distortion resulting from a relatively rare 3-jus delay-
spread environment is still more then 18 dB below the speech.

Both equal gain and maximal ratio space diversity can be applied to
the previously described ssb receiver. The latter would be expected to
have a slight edge in performance improvement. Equal gain diversity
was selected for this analysis for convenience. Two-branch, equal gain
diversity was seen to significantly mitigate the previously described
impairments. The amount of distortion, and the s/i disadvantage for
a given gain limit, is considerably reduced. In addition, the use of
unlimited correction gain is feasible without adverse effects on inter-
ference and delay spread performance. In fact, correction gain in a
diversity receiver would probably be determined by the diminished
returns in audio quality and by hardware considerations. This point is
likely to be found somewhere between 10 and 20 dB of correction. A
two-branch diversity receiver with 20 dB of correction would have
distortion that falls 45 dB below the voice in a nondelay spread
environment, and more than 24 dB below the voice when the delay
spread is as high as 3 /xs. The s/i disadvantage for this case would be
less than 1 dB.

SSB VOICE SIGNAL IMPAIRMENT MECHANISMS 1941

REFERENCES

1. A. L. Hopper, "An Experimental Fast Acting AGC Circuit," IRE Int. Conv. Rec, 10,

Part 8 (March 1962), pp. 13-20.

2. W. C. Jakes, ed., Microwave Mobile Communications, New York: Wiley, 1974.

3. UHF Task Force to the FCC, "Spectrum-Efficient Technology for Voice Commu-

nications," Report of the UHF Task Force, Office of Plans and Policy, Federal
Communications Commission, Washington D.C., Feb. 1978.

4. B. Lusignan, "Single-Sideband Transmission for Land Mobile Radio," IEEE Spec-

trum, 15 (March 1978), pp 33-37.

5. R. W. Gibson and R. Wells, "The Potential of ssb For Land Mobile Radio," Rec.

29th IEEE-VT Conf. (March 1979).

6. W. Gosling, "Single Sideband as a Contribution to Spectrum Efficient Civil Land

Mobile Radio," submission to the fcc on Recent Mobile Radio Research from the
University of Bath, School of Electrical Engineering, Feb. 1979.

7. R. Freudberg, "A Laboratory Simulator for Frequency Selective Fading," IEEE

First Ann. Commun. Conf. (1965), pp. 609-614.

8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, AMS-55,

Washington, D.C.: National Bureau of Standards, June 1964.

9. D. C. Cox, "Multipath Delay Spread and Path Loss Correlation for 910 MHz Urban

Mobile Radio Propagation," IEEE Trans. Veh. Technol., VT-23, No. 4 (Nov.
1977), pp. 340-344.

1 942 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1 980